Desynchronization with an Artificial Force Field for Wireless Networks

Transcription

1 Desynchronization with an Artificial Force Field for Wireless Networks Supasate Choochaisri Kittipat Apicharttrisorn Kittiporn Korprasertthaworn Pongpakdi Taechalertpaisarn Chalermek Intanagonwiwat Department of Computer Engineering Chulalongkorn University Bangkok, Thailand {supasatec, kittipatap, kittipornk, ABSTRACT Desynchronization is useful for scheduling nodes to perform tasks at different time This property is desirable for resource sharing, TDMA scheduling, and collision avoiding Inspired by robotic circular formation, we propose DWARF (Desynchronization With an ARtificial Force field), a novel technique for desynchronization in wireless networks Each neighboring node has artificial forces to repel other nodes to perform tasks at different time phases Nodes with closer time phases have stronger forces to repel each other in the time domain Each node adjusts its time phase proportionally to its received forces Once the received forces are balanced, nodes are desynchronized We evaluate our implementation of DWARF on TOSSIM, a simulator for wireless sensor networks The simulation results indicate that DWARF incurs significantly lower desynchronization error and scales much better than existing approaches Categories and Subject Descriptors C [Computer-Communication Networks]: Network Architecture and Design Distributed Networks General Terms Algorithms, Design, Performance Keywords desynchronization, self-organizing, sensor networks, wireless networks INTRODUCTION Networked, distributed systems usually cooperate to have a common notion of time in order to accomplish tasks with consistent results through time synchronization protocols Authors would like to thank Assoc Prof Dr Paisan Nakmahachalasint and three anonymous reviewers for their valuable comments and suggestions to revise and improve this work Supported by CU CP Academic Excellence Scholarship from Department of Computer Engineering, Faculty of Engineering, Chulalongkorn University Corresponding Author Figure : Desynchronization Framework (Left) Nodes are placed and labeled on the time circle Node i has the time phase φ i and the phase difference i with its previous phase neighbor i+ is the phase difference between node i and its next phase neighbor (Right) The positions of nodes in the perfect desynchrony state (eg, TPSN [5], FTSP [9], GTSP [6], and EGTSP []) However, some systems simply require nodes to work at the same time (eg, Firefly Synchronicity [7]) In such systems, a global notion of time may not be necessary Conversely, some systems require nodes not to work at the same time (ie, to desynchronize) Desynchronization organizes all accesses to a shared resource to be collision-free and even equitable A concrete example is a system using a Time Division Multiple Access (TDMA) protocol Nodes access the shared media only in their time slots to send messages with no collision Other potential applications are techniques to increase a sampling rate in multiple analog-to-digital converters, to schedule resource in multi-core processors, and to control traffic at intersections DESYNC [4] proposes a simple desynchronization framework as depicted in Figure The perimeter of a time circle represents a configurable firing period T of nodes oscillators The time position or phase of each node represents its turn to access a shared resource The system is desynchronized when all nodes are equally separated in the time circle Desynchronization can be categorized into two groups: Weak desynchronization(eg, INVERSE-MS[3]) distributes nodes equally in the time circle but distorts the time period (ie, the period is extended) Therefore, it is not suitable for systems that require an exact time period to access resources ACM SIGCOMM Computer Communication Review 8 Volume 4, Number, April

2 Strong desynchronization (eg, DESYNC [4], M-DE- SYNC [8]) is more desirable than weak desynchronization because of no period distortion Similar to DESYNC, we focus only on strong desynchronization due to their mentioned desirable property The DESYNC algorithm is simple and effective The phase of each node is an average phase of its two phase neighbors on the time circle However, DESYNC s error is quite high even after convergence because a phase error of a node can propagate to its phase neighbors and indefinitely circulate in the network (see Section 6) In this paper, we present DWARF (Desynchronization With an ARtificial Force field), a novel desychronization protocol that incurs low desynchronization error even in a highly dense network Our work is inspired by the robot circular formation with an electromagnetic-like force field [] In that work, robots of the same type have artificial forces to repel each other whereas robots of the different types have attracting forces For a certain coefficient of forces, a circle of heterogeneous robots can be formed (see Figure ) Similarly, if we think of nodes on a time circle as the robots of the same type, the nodes will repel each other and keep time intervals from their neighbors as far as possible Once all received forces are balanced, nodes are equally spread out in the time circle (ie, desynchronized) DWARF have the following contributions: DWARF is a distributed desynchronization algorithm using time phases of all neighbors to achieve the desyncrhony state DWARF does not require time synchronization, does not assume already slotted time, and does not incur any control message overhead DWARF is simple due to low complexity in terms of computation and memory Therefore, it is suitable for resource-constraint networks, such as wireless sensor networks Additionally, message complexity is low because the algorithm relies on the timing of the message, not information inside the message We have implemented and evaluated DWARF on TOS- SIM, a simulator for wireless sensor networks Our results indicate that DWARF scales well with network size and outperforms DESYNC significantly by achieving - 63% reduction in desynchronization error The rest of the paper is organized as follows Section covers the related works about desynchronization in wireless networks In Section 3, we overview the robot circular formation that motivates our work We describe our desynchronization protocol in Section 4 Section 5 analyses the convergence property of the protocol Then, we evaluate the performance in Section 6 Finally, Section 7 concludes the paper and discusses limitations as well as future works RELATED WORK To the best of our knowledge, DESYNC [4] is the first to introduce the desynchronization problem In DESYNC, a node simply attempts to stay in the middle between its previous and next phase neighbors By repeating this simple algorithm, all nodes will eventually be spread out However, the error is also propagated to the phase neighbors and indefinitely circulated inside a network Therefore, the DESYNC s error is quite high even after convergence Using phases of all neighbors, DWARF can achieve significantly lower desynchronization error than that of DESYNC Designed to converge faster than DESYNC, INVERSE- MS [3] is an inverse algorithm of the synchronicity work by Morollo and Strogatz [] At a steady state, INVERSE-MS maintains a dynamic equilibrium (ie, nodes keep changing time phases while maintaining desynchronization) However, the time period is distorted (ie, weak desynchronizaiton) M-DESYNC [8] proposes a localized multi-hop desynchronization protocol that works on a granularity of time slots M-DESYNC is strong desynchronization protocol that is designed to work on a multi-hop network The protocol estimatestherequirednumberoftimeslotswithatwo-hopmaximum degree This estimation helps M-DESYNC converge very fast However, M-DESYNC requires that all nodes have a global notion of time in order to share the common perception of time slots Additionally, M-DESYNC also requires additional message overhead for claiming the slots Conversely, DWARF does not require a global notion of time or additional message overhead Neither does DESYNC Motskin et al [] proposes a simple and lightweight strong desynchronization algorithm that is based on a graph coloring model The lightweight coloring algorithm works on general graph networks and does not need global time To ensure that the selected time slot does not overlap with others, a node needs to listen to the media for a full time period before claiming the slot Without a common notion of time, the starting time of each slot is quite random As a result, several time gaps are too small to be used as time slots This external fragmentation problem reduces resource utilization of the system Additionally, to converge faster, their algorithm overestimates the number of time slots Hence, several large time gaps are also left unused and the resource utilization is low In contrast, DESYNC and DWARF fully utilize resources even without a global notion of time Recently, Degesys and Nagpal[3] extend DESYNC to support multi-hop networks We also plan to similarly extend DWARF for multi-hop networks in our future work as well Other works that are similar to desynchronization protocols are distributed TDMA protocols Most of distributed TDMA protocols (eg, TRAMA [4], Parthasarathy [], ED-TDMA [6], and Herman [7]) assume time is already slotted or all nodes are synchronized to achieve the same global clock Some distributed TDMA protocols do not require time synchronization However, they require more states and incur control message overhead For example, DRAND [5] requires the control overhead for sending request, reject, release, and grant messages In our work, again, we do not require time synchronization, do not assume already slotted time, and do not incur any control message overhead 3 MOTIVATION We observe that the desynchronization framework is similar to the circular formation of heterogeneous robots [] In robotics, some tasks are too difficult for a single robot to accomplish by itself Some even require multiple robots with different types For such tasks, heterogeneous robots are distributedly grouped into teams that are equally spread out to cover the monitored area Each robot has no global knowledge of others absolute positions but can detect rela- ACM SIGCOMM Computer Communication Review 9 Volume 4, Number, April

3 Figure : Results of Robotic Circular Formation in [] Robots with two different types form the circle tivepositionsoftheotherswithrespecttoitselfaswellasthe type of the others To form a circle, an artificial force is used as an abstraction for velocity adaptation of a robot Robots of different types have attracting forces to each other Conversely, robots of the same type have repelling forces As a result, the circle of heterogeneous robots will be formed and robots are nicely spaced on the circle (see Figure ) In our work, a node (like a robot) does not have a global notion of time but each node can measure relative time differences with other nodes The circle of robots is similar to our circle of time period The distribution of robots can be mapped to the distribution of nodes on the time circle The work inspires us to design a novel desynchronization protocol for wireless networks based on an artificial force field In this paper, the circle is temporal rather than spatial and only repelling forces are required to space nodes on the time circle 4 DESYNCHRONIZATION PROTOCOL We first describe the concept of an artificial force field in Section 4 and explain our algorithm in Section 4 Like DESYNC [4] and INVERSE-MS [3], we assume a one-hop network in this paper Similar to DESYNC, our protocol can also be extended for multi-hop networks We discuss this assumption in Section 7 4 Artificial Force Field An artificial force field is an analogy to the circle of a time period Nodes are in the same force field if they can communicate with each other If node i and node j are on the same force field, they have repelling forces to push one another away A closer pair of nodes has a higher magnitude of force than a farther pair does The time interval between two nodes is derived from the phase difference between them If two nodes have a small phase difference, they have a high magnitude of force and vice versa In other word, a repelling force is an inverse of a phase difference between two nodes: f ij = φ ij/t, φij ( T, T ), () where f ij is the repelling force from node j to node i on a time period T and φ ij is the phase difference between node i and j We note that φ ij is not equal to because if two nodes fire at the same time, their firings collide and two nodes do not record other s firing Additionally, at T or T, a node does not repel an opposite node because they are balanced A repelling force can be positive (clockwise repelling) or negative (counterclockwise repelling) A positive force is created by a node on the left half of the circle relative to Figure 3: Artificial Force Field Arrow lines represent repelling forces from node, 3, and 4 to node A shorter and thicker line is a stronger force A force from node 4 is a positive force and two forces from node and 3 are negative forces the considered node whereas a negative force is created by a nodeontherighthalf Figure3representsafieldofrepelling forces on node Each node in the force field moves to a new time position or phase proportional to the total received force Given n nodes in a force field, the total force on a node i is the following: F i = f ij () j= Eventually, nodes reach an equilibrium state whereby the total force of the system is close to zero and each pair of phase neighboring nodes has the same time interval This equilibrium state also indicates the desynchrony state because all nodes are equally spaced on the time circle 4 Algorithm We assume that, initially, nodes are not desynchronized Each node sets a timer to fire in T time unit After setting the timer, each node listens to all neighbors until its timer expires When receiving a firing message from its neighbor, the (positive or negative) repelling force from that neighbor is calculated based on the phase difference When the timer expires, a node broadcasts a firing message to neighbors Then, the node calculates a new time phase to move on the circle based on the summation of forces from all neighbors and sets a new timer according to the new time phase Reasonably, one may wonder how far a node should move or adjust its phase In our work, given the total received force F i, the node i adjusts to a new time phase φ i, φ i = (φ i +KF i) mod T, (3) where φ i is the current phase of the node i Undoubtedly, the proper value of the coefficient K leads to the proper new phase The value of K is similar to a step size which is used in artificial intelligence techniques Therefore, if the value of K is too small, the system takes much time to converge On the other hand, if the value of K is too large, the system may overshoot the optimal value and does not converge We observe that, given the same time period, fewer nodes in the system result in bigger phase difference between two phase neighbors To be desynchronized, nodes in sparse networks must make a bigger adjustment to their time phases than nodes in dense networks must Therefore, the same total received force should have a bigger impact on ACM 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4 K 5 5 Data Trendline Number of Neighbors (nodes) Figure 4: Relation of the coefficient K with the number of nodes n a node in sparse networks than on a node in dense networks To reflect this observation, the coefficient K is inversely proportional to a power of the number of nodes n, K = c n c,c,c (4) Therefore, we have conducted an experiment to find the proper value of c and c We set the time period T to Additionally, we have varied the number of nodes and have searched for the K values that led to low errors (see Figure 4) Using power regression on the result, we have deduced a relation of K and n (the trendline) We have also found the proper value of c and c as follows: K = n 874 However, this K value is derived by setting T equal to Therefore, for arbitrary T, K = n 874 T (5) We explain other details of the experiment and proof of Equation 5 in Appendix A All nodes in the artificial force field (in the period circle) iteratively run the same algorithm until the forces are balanced (ie, all nodes are in the desynchrony state) The pseudo-code of this algorithm is shown in Appendix B 5 STABILITY ANALYSIS To mathematically analyze the stability of the algorithm (ie, the algorithm converges to the steady desynchronized state) is rather complicated The stability analysis of standard dynamical systems does not suffice because the force function is non-linear and the transformation matrix to find the eigenvalues cannot be formed Therefore, we have tested the stability by simulation In each simulation scenario, nodes randomly wake up Our simulation result shows that the system always converge except when there are too many nodes within a short period (ie, the time gap between each firing may be shorter than the message delay) In other words, the system is over-saturated The result is shown in Section 63 We believe that the non-linear dynamic systems analysis based on the Lyapunov stability theory could prove our conjecture However, one of the reasons that the system converges (nodes are nicely spread) is that our objective function (ie, the summation of received forces at all nodes) is convex Therefore, it contains only one global minima and no local minima The proof of convexity is shown in Appendix Normalized Root Mean Square Error 5 DESYNC DWARF Number of Neighbors (nodes) Figure 5: Root mean square error normalized by perfect phase difference after 3 time periods D Our algorithm attempts to reduce the value of the objective function overtime and eventually reach a value near the global minima However, the system does not always converge To converge, the system must meet two criteria First, the value of K must be proper (see Section 4) Second, the number of nodes within a time period must not be too high Otherwise, the system may be over-saturated 6 EVALUATION In this section, we evaluate the performance of our proposed mechanism and compare with DESYNC [4] because DESYNC and our mechanism share the same goal and requirements Particularly, they do not require time synchronization, do not assume already slotted time, do not need to look into the packet content, and do not incur control packets but still achieve equivalent time spaces Other protocols (eg, M-DESYNC [8], Lightweight []) assume different requirements Therefore, we only discuss our differences with such protocols in Section The performance metrics in this evaluation are desynchronization error and convergence time The former indicates how close the current state is to the perfect desynchrony state The latter indicates how fast the algorithm converges 6 Evaluation Environment We implement DWARF on TinyOS, an operating system for wireless sensor networks and evaluate the protocol on TOSSIM, a TinyOS simulator We vary the one-hop network size from 4 to 64 nodes Each node periodically fires a message that contains only application data with no extra control overhead This zero overhead is the advantage of both DWARF and DESYNC because, to avoid collisions, a node only needs to know the timing of the firing rather than the control information inside a packet In our simulation, for both DWARF and DESYNC, we use a -byte node ID and a -byte counter as the data However, we do use the regular -byte CC4 header for TOSSIM Therefore, we do not measure the overhead in our evaluation We set the time period to 5 milliseconds and compare our result with that of DESYNC The step size (α) of DESYNC is set to 95 (the same value used in [4]) Initially, the phase of each node is random in a range of to 5 6 Desynchronization Error To measure the desynchronization error, we run the simulation for 3 time periods In each network size, we run the simulation for 3 times Then, we measure the average root mean square error (RMSE) The error (ERR) is the mea- ACM SIGCOMM Computer Communication Review Volume 4, Number, April

5 Normalized Root Mean Square Error 5 DESYNC-8 DWARF-8 DESYNC-4 DWARF (a) Sparse (4 and 8 nodes) Normalized Root Mean Square Error 5 DESYNC-3 DWARF-3 DESYNC-6 DWARF (b) Dense (6 and 3 nodes) Normalized Root Mean Square Error 5 DESYNC-64 DWARF-64 DESYNC-48 DWARF (c) Extremely Dense (48 and 64 nodes) Figure 6: Convergence time and root mean square error normalized by expected phase difference sured phase difference minus the perfect phase difference: ERR i = φ ij T/n, n i= RMSE = ERR i, n where node j is the next phase neighbor of node i φ ij is the phase difference between node i and node j on the time period T Given that n is the total number of nodes, T/n is the perfect phase difference However, a smaller absolute error in a dense network is not necessarily better than a bigger error in a sparse network because the perfect phase difference in a dense network is also smaller than that in a sparse network Thus, for a comparable view of each network size, we also measure a normalized root mean square error (NRMSE) that is a ratio of the root mean square error and the perfect phase difference of each network size Figure 5 illustrates the result of the normalized desynchronization error in each network size after 3 time periods (see Figure C in Appendix C for the absolute root mean square errors) The result indicates that, in all network sizes (4-64 nodes), DWARF achieves significantly better desynchrony states than DESYNC does Understandably, using information from all neighbors (as in DWARF) leads to lower errors than using information from only two phase neighbors (as in DESYNC) does Furthermore, DESYNC s mechanism allows the phase error of a node to propagate to its phase neighbors A part of this error will propagate back and forth between two phase neighbors as well as circulate inside the network As a result, DESYNC s error after convergence is still quite large In contrast, DWARF is robust to this error propagation Even though the error propagation may still occur, the impact is not significant Given that DWARF uses the sum of forces from all neighbors, an error from one neighbor does not overwhelm the system We note that we show RMSE and NRMSE after 3 periods because, by that time, the errors of all simulation scenarios seem to be stable The actual convergence time in most scenarios is much lower than 3 rounds (see the next section) 63 Convergence Time In desynchronization error evaluation, we only measure the performance after 3 time periods However, the previous result does not indicate whether the protocols have converged or not Neither does it indicate how fast the protocols have converged Hence, we also measure the absolute root mean square error and normalized root mean square error for each time period (absolute root mean square errors are shown in Appendix C) In sparse networks (Figure 6a), both protocols converge with similar small error but DWARF converges slightly faster In dense networks(figure 6b) and extremely dense networks (Figure 6c), DESYNC converges but errors highly fluctuate whereas DWARF converges faster and errors are more stable Furthermore, in dense and extremely dense networks, the normalized root mean square errors of DESYNC after convergence is higher or equal to This means that the error is very large compared to the perfect phase difference Conversely, the normalized error of DWARF is lower than even in the extremely dense networks Therefore, DWARF scales well with the network density whereas DESYNC does not In a denser network, the errors of DESYNC and DWARF are higher because the probability of message collisions increases (see the next section) 64 Correlation of Packet Loss and Desynchronization Error In this section, we investigate the correlation of packet loss and desynchronization error by setting all nodes to wake up simultaneously Due to space limitation, we only include the results of DWARF in networks of 8, 3, and 64 nodes for 3 time periods (Figure 7) However, in networks of 4, 6, and 48 nodes, the results are similar (not shown) At the beginning, the network is far from the desynchrony state and messages from different nodes are simultaneously fired This results in lost packets and errors However, over time, nodes gradually adapt their time phases Consequently, the number of lost packets and the error also gradually drop 7 CONCLUSION AND DISCUSSION In this paper, we present DWARF, a novel strong desynchronization protocol that enables nodes in a system to perform tasks at different time To the best of our knowledge, DWARF is the first desynchronization protocol that is based on the concept of electromagnetic fields, a foundation of physics Our protocol is completely distributed and localized with no reliance on a centralized node or a global notion of time Due to low complexity in terms of computation, memory, and message overhead, DWARF is suitable for traditional ACM SIGCOMM Computer Communication Review Volume 4, Number, April

6 Normalized root mean square error NRMSE Normalized root mean square error NRMSE Normalized root mean square error NRMSE (a) 8 nodes (b) 3 nodes (c) 64 nodes Figure 7: Correlation of packet loss and desynchronization error wireless networks as well as resource-constraint wireless sensor networks Our preliminary result indicates that DWARF can significantly outperform DESYNC by reducing - 63% of the desynchronization error In addition, DWARF scales well with network size given that the normalized error is lower than even in extremely dense networks Giventhatthisworkisstillinanearlystage, thereisroom to improve and to extend the idea before we can realize the full potential of DWARF Recently, Degesys[3] has extended the DESYNC concept to support multi-hop topologies The extension is also applicable to DWARF Therefore, we believe that DWARF can be similarly extended for multi-hop topologies as well In addition, the force function of DWARF is not limited to the sum of rational functions Other different force functions (eg squared, cubed) are possible and viable for further investigation We plan to evaluate DWARF on real wireless devices and to explore other extensions in our future work 8 REFERENCES [] K Apicharttrisorn, S Choochaisri, and C Intanagonwiwat Energy-efficient gradient time synchronization for wireless sensor networks In Proc of the International Conference on Computational Intelligence, Communication Systems and Networks, CICSyN, [] N Boonpinon and A Sudsang Heterogeneity driven circular formation In Proc of the IEEE International Conference on Robotics and Biomimetics, ROBIO 6, 6 [3] J Degesys and R Nagpal Towards desynchronization of multi-hop topologies In Proc of the IEEE International Conference on Self-Adaptive and Self-Organizing Systems, SASO 8, 8 [4] J Degesys, I Rose, A Patel, and R Nagpal DESYNC: Self-organizing desynchronization and TDMA on wireless sensor networks In Proc of the International Symposium on Information Processing in Sensor Networks, IPSN 7, 7 [5] S Ganeriwal, R Kumar, and M B Srivastava Timing-sync protocol for sensor networks In Proc of the International Conference on Embedded Networked Sensor Systems, SenSys 3 ACM, 3 [6] H Gong, M Liu, G Chen, and X Zhang A study on event-driven tdma protocol for wireless sensor networks EURASIP J Wirel Commun Netw, :9: 9:, February [7] T Herman and S Tixeuil A distributed tdma slot assignment algorithm for wireless sensor networks In Algorithmic Aspects of Wireless Sensor Networks, volume 3 of Lecture Notes in Computer Science, pages Springer Berlin / Heidelberg, 4 [8] H Kang and J Wong A localized multi-hop desynchronization algorithm for wireless sensor networks In Proc of the IEEE INFOCOM, 9 [9] M Maróti, B Kusy, G Simon, and A Lédeczi The flooding time synchronization protocol In Proc of the International Conference on Embedded Networked Sensor Systems, SenSys 4 ACM, 4 [] R E Mirollo and S H Strogatz Synchronization of pulse-coupled biological oscillators SIAM J Appl Math, 5:645 66, November 99 [] A Motskin, T Roughgarden, P Skraba, and L Guibas Lightweight coloring and desynchronization for networks In Proc of the IEEE INFOCOM, 9 [] S Parthasarathy and R Gandhi Distributed algorithms for coloring and domination in wireless adhoc networks In FSTTCS 4: Foundations of Software Technology and Theoretical Computer Science, volume 338 of Lecture Notes in Computer Science, pages [3] A Patel, J Degesys, and R Nagpal Desynchronization: The theory of self-organizing algorithms for round-robin scheduling In Proc of the International Conference on Self-Adaptive and Self-Organizing Systems, SASO 7, 7 [4] V Rajendran, K Obraczka, and J J Garcia-Luna-Aceves Energy-efficient, collision-free medium access control for wireless sensor networks Wirel Netw, :63 78, February 6 [5] I Rhee, A Warrier, J Min, and L Xu Drand: Distributed randomized tdma scheduling for wireless ad hoc networks IEEE Transactions on Mobile Computing, 8(): , oct 9 [6] P Sommer and R Wattenhofer Gradient clock synchronization in wireless sensor networks In Proc of the International Conference on Information Processing in Sensor Networks, IPSN 9, 9 [7] G Werner-Allen, G Tewari, A Patel, M Welsh, and R Nagpal Firefly-inspired sensor network synchronicity with realistic radio effects In Proc of the International Conference on Embedded Networked Sensor Systems, SenSys 5 ACM, 5 ACM SIGCOMM Computer Communication Review 3 Volume 4, Number, April

7 APPENDIX A FINDING K-N RELATION In this section, we explain in details how we find the proper values of c and c for the relation between K and n values We set a time period T to and vary the number of nodes In the specific number of nodes, we first simulate to see the trend of the value K that leads to small errors Then, we select a range of good K values After that, we simulate times to obtain the average desynchronization error for each K value In each simulation, we randomly set aninitialphaseofeachnodebetweenandt (periodvalue) Finally, weselectthek valuethatresultsinthelowesterror After getting the proper K value for each number of nodes, we plot the relation between K and the number of nodes (Figure 4) and use a mathematic tool to calculate the power regression The obtained relation function between K and n (the trendline in Figure 4) consists of c and c values as follows: K = n 874 However, this K value is derived by setting T equal to Therefore, for arbitrary T, K = n 874 T Proof From Equation and 3, the phase of node j is adjusted by KF j = K n i j fi,j = n i j Kfi,j Therefore, we can analyze the value of K from only single force f i,j For the time circle of a period T p, let θ p be an angle between two nodes on the circle and Θ p be an angle between the new adjusted phase and the old phase based on a single force where θ p,θ p (,π) Hence, and θ p π = φi,j T p, (6) Θ p π = Kfi,j T p (7) Ifθ oft equalstoθ oft, bothofthemshouldbeadjusted with the same angle amount Θ = Θ Thus, from Equation 7, K f i,j() T Θ = Θ From Equation and 6, f i,j = = Kf i,j() T K π = Kπ T θ T θ K = K T T φ i,j /T p = π θ p, consequently, K = K T T (8) At T =, we get K = n 874 Therefore, from Equation 8, for arbitrary T, K = n 874 T Root Mean Square Error DESYNC DWARF Number of Neighbors (nodes) Figure C: Absolute root mean square error after 3 time periods B PSEUDOCODE : Initialization : T = T imep eriod {Configurable Time Period} 3: n = {Number of receiving messages including itself} 4: F = {Force Summation} 5: lastf iringt ime = localt ime 6: currentp hase = localt ime modulo T 7: Set a firing timer to be T unit time 8: Upon timer firing 9: Broadcast a firing message to neighbors : lastf iringt ime = localt ime : currentp hase = localt ime modulo T : K = n 874 T 3: newphase = currentphase+(k F) 4: if newphase < then 5: newphase = T +newphase 6: end if 7: Set a firing timer to be fired at (newphase modulo T) 8: F = 9: n = : Upon receiving a firing message : n = n+ : phasediff = localt ime lastf iringt ime 3: if phasediff == T then 4: F = F + {Balanced force} 5: else if phasediff < T then 6: F = F + {Positive force} phasediff/t 7: else 8: F = F {Negative force} (T phasediff)/t 9: end if C ABSOLUTE ROOT MEAN SQUARE ER- ROR In this section, we show the simulation results of average root mean square errors without normalization Figure C indicates the absolute root mean square errors after 3 rounds that we evaluate in Section 6 Figure C indicates the convergence time and the absolute root mean square errors that we evaluate in Section 63 D PROOF OF CONVEXITY Theorem The system force summation function has one global minima and no local minima ACM SIGCOMM Computer Communication Review 4 Volume 4, Number, April

8 Root Mean Square Error DESYNC-8 DWARF-8 DESYNC-4 DWARF-4 Root Mean Square Error DESYNC-3 DWARF-3 DESYNC-6 DWARF-6 Root Mean Square Error DESYNC-64 DWARF-64 DESYNC-48 DWARF (a) Sparse (b) Dense (c) Extremely Dense Figure C: Convergence time and absolute root mean square error To prove Theorem, let F i be the force summation at node i and i,j ( T, T ) is an interval between node i and j If i,j >, the node j repels the node i in a positive direction In contrast, if i,j <, the node j repels the node i in a negative direction Therefore, for n nodes, F i can be formulated as the following equation, F i = i,j j= The objective function E of the system is the summation of absolute received forces at all nodes, E = F i = i= i= i,j (9) j= To prove the function E is a convex function, we must prove that two following conditions are satisfied; ) the set of all i,j isaconvexsetand)thehessianmatrixofe ispositive semidefinite Proposition A set of all possible interval i,j is a convex set Proof A set of ( T, T ) is a line connecting between T and T Therefore, any i,j, i,j ( T, T ) and α R with α, α i,j +( α) i,j ( T, T ) Proposition The Hessian of the function E is positive semidefinite Proof Let H be the Hessian matrix of the objective function E, E E,, n,n E H =,3, E,3 n,n E n,n, E n,n For any x,y, we derive the first-order derivative as follows, E x,y = nj= j x nj= j x x,j x,j ( ) x,y For the second-order partial derivatives, if we differentiate with the same x,y, E = x,y nj= j x nj= j x x,j x,j ( 3 x,y ) In the other hand, if we differentiate with other u,v, where u x or v y, u,v x,y E = Therefore, the Hessian matrix of the function E is a, a,3 H =, a ( ) n,n where a i,j = u i u i 3 i,j and u i = n j= i,j To show that the Hessian matrix of the function E is positive semidefinite, we show that T H for all when T H = ( ),,3 n,n H = = = i= i= j= u i u i ( ) u i u i i,j j= i,j n u i u i= i ui = i= u i,,3 n,n Therefore, we can conclude that T H and the Hessian matrix of E is positive semidefinite From Proposition and, we derive the following lemma Lemma the function E is a convex function From Lemma, we consequently prove Theorem that the system force summation function has one global minima and no local minima ACM SIGCOMM Computer Communication Review 5 Volume 4, Number, April